Explicit estimates for the Stirling numbers of the second kind

TL;DR

This paper provides explicit, asymptotically sharp estimates for Stirling numbers of the second kind, using probabilistic representations that vary depending on the region of m within the set {1,...,n}.

Contribution

It introduces new explicit estimates for S(n,m) that are asymptotically sharp, employing different probabilistic methods for central and non-central m.

Findings

Estimates vary according to the position of m within {1,...,n}

Most estimates are asymptotically sharp with few exceptions

Different probabilistic representations are used for different regions

Abstract

We give explicit estimates for the Stirling numbers of the second kind $S(n,m)$. With a few exceptions, such estimates are asymptotically sharp. The form of these estimates varies according to $m$ lying in the central or non-central regions of $\{1,\ldots ,n\}$. In each case, we use a different probabilistic representation of $S(n,m)$ in terms of well known random variables to show the corresponding results.